Higher resonance schemes and Koszul modules of simplicial complexes (Q6570587)

Working over a characteristic zero base field $\mathbf{k}$, in the article under review, the authors study the higher Koszul modules $W_i(\Delta)$ that are associated to each abstract simplicial complex $\Delta$ on $n$ vertices.\N\NIn more precise terms, a construction of Stanley and Reisner associates to $\Delta$ a graded, graded-commutative algebra $\mathbf{k}\langle \Delta \rangle$; denoting by $S$ the polynomial ring $\mathbf{k}[x_1,\dots,x_n]$, endowed with its standard multigrading, there is a multigraded chain complex $(\mathbf{k}\langle \Delta \rangle_\bullet \otimes_{\mathbf{k}} S, \partial_\bullet)$ and the higher Koszul modules $W_i(\Delta)$ are defined to be the multigraded $S$-modules \N\[\NW_i(\Delta) := H_i (( \mathbf{k} \langle \Delta \rangle_\bullet \otimes_{\mathbf{k}} S , \partial_\bullet) ).\N\]\NThe authors' first result is that for all $i \geq 1$ and each square-free multi-index $\mathbf{b} \in \mathbb{N}^n$, there is a natural isomorphism of vector spaces \N\[\N[W_i(\Delta)]_{\mathbf{b}} \simeq [\operatorname{Tor}^S_{|\mathbf{b}| - i}(\mathbf{k},\mathbf{k}[\Delta])]_{\mathbf{b}}^\vee.\N\]\NHere, $\mathbf{k}[\Delta]$ is the polynomial Stanley-Reisner ring of $\Delta$.\N\NSecond, the authors show that the multigraded Hilbert series of $W_i(\Delta)$ can be described as \N\[\N\sum_{\mathbf{a} \in \mathbb{N}^n} \operatorname{dim}_{\mathbf{k}}[W_i(\Delta)]_{\mathbf{a}} \mathbf{t}^{\mathbf{a}} = \sum_{\substack{ \mathbf{b} \in \mathbb{N}^n \\\N\mathbf{b}\text{ is square-free}}} \operatorname{dim}_{\mathbf{k}}(\tilde{H}_{i-1}(\Delta_{\mathbf{b}};\mathbf{k})) \frac{ \mathbf{t}^{\mathbf{b}}}{\prod_{j \in \operatorname{Supp}(\mathbf{b})}(1-t_j)}.\N\]\NHere, $\Delta_{\mathbf{b}}$ is the restriction of $\Delta$ to $\operatorname{Supp}(\mathbf{b})$ and $\tilde{H}_{i-1}(\Delta_{\mathbf{b}};\mathbf{k})$ is the $(i-1)$th reduced homology group of $\Delta_{\mathbf{b}}$ with coefficients in $\mathbf{k}$.\N\NFinally, the authors prove that for each $i \geq 1$, the scheme structure on the support resonance loci \N\[\N\mathcal{R}_i(\Delta) := V(\operatorname{Ann}(W_i(\Delta)))\N\]\Nis reduced. Further, they give a description of its decomposition into irreducible components.\N\NThe results of the present article complement those of the companion article [\textit{M. Aprodu} et al., ``The second syzygy schemes of curves of large degree'', Preprint, \url{arXiv:2409.11855}].